Properties

Label 1148.169
Modulus $1148$
Conductor $41$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1148, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,11]))
 
pari: [g,chi] = znchar(Mod(169,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.bn

\(\chi_{1148}(169,\cdot)\) \(\chi_{1148}(197,\cdot)\) \(\chi_{1148}(225,\cdot)\) \(\chi_{1148}(449,\cdot)\) \(\chi_{1148}(617,\cdot)\) \(\chi_{1148}(841,\cdot)\) \(\chi_{1148}(869,\cdot)\) \(\chi_{1148}(897,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{41})^+\)

Values on generators

\((575,493,785)\) → \((1,1,e\left(\frac{11}{20}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(i\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
value at e.g. 2